Commutativity and non-commutativity of topological sequence entropy

Balibrea, Francisco; Peña, Jose Salvador Cánovas; López, Víctor Jiménez

Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1693-1709.

Dans cet article nous étudions la propriété de commutativité pour l’entropie séquentielle topologique. Nous prouvons que si $X$ est un espace métrique compact et $f,g:X\to X$ sont deux fonctions continues, alors $h_A(f\circ g)=h_A(g\circ f)$ pour toute suite croissance $A$ où $X=[0,1]$ et nous construisons un contre-exemple dans le cas général. Au passage, nous prouvons aussi que l’égalité $h_A\left(f\right)=h_A(f{|}_{{\cap }_{n\ge 0}{f}^n\left(X\right)})$ est vraie si $X=[0,1]$ mais ne l’est pas nécessairement si $X$ est un espace métrique compact arbitraire.

In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g:X\to X$ are continuous maps then $h_A(f\circ g)=h_A(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality $h_A\left(f\right)=h_A(f{|}_{{\cap }_{n\ge 0}{f}^n\left(X\right)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

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